Logic Encodings in LF: A Completeness Criterion Florian Rabe Jacobs - - PowerPoint PPT Presentation

Logic Encodings in LF: A Completeness Criterion

Florian Rabe

Jacobs University Bremen

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LF

◮ LF = A Framework for Defining Logics (Harper, Honsell,

Plotkin; 1993)

◮ Impredicative dependent type theory, related to Martin-L¨

• f

type theory

◮ Curry-Howard equivalent to first-order logic with predicates,

implication and universal quantifier

◮ Types:

◮ Application of type-valued constant, e.g.,

if Matrix:N → N → type, then Matrix(5, 4):type

◮ Dependent product, e.g.,

if I : Π x:N. Matrix(x, x), then I(3):Matrix(3, 3) (e.g., identity matrix)

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Logic Encodings

◮ LF very suitable for logic encodings (esspecially natural

deduction or sequent calculus)

◮ Example: Fragment of propositional logic with natural

deduction form:type ∧:form → form → form proof:form → type ∧I :proof(F) → proof(G) → proof(F ∧ G) ∧El :proof(F ∧ G) → proof(F) ∧Er :proof(F ∧ G) → proof(G)

◮ Structural rules (axiom, weakening, exchange) naturally

derivable in the type theory

◮ Model theory not covered

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LF as a Logic

◮ Proof and model theory for LF developed building on joint

work with Steve Awodey

◮ Permits to

◮ encode model theory of logics as well as proof theory ◮ formalize encodings as institution translation into LF ◮ reason about logic encodings

◮ Formulas: equalities for all types, first-order connectives,

first-order quantifiers for all types, classical negation

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Proof theory of the LF meta-logic (examples)

F ⊢Σ F ′ ⊢Σ F ⇒ F ′ ⊢Σ F ⇒ F ′ ⊢Σ F ⊢Σ F ′ ⊢Σ F ⊢Σ F ′ ⊢Σ F ∧ F ′ ⊢Σ F ∧ F ′ ⊢Σ F ⊢Σ F ∧ F ′ ⊢Σ F ′ x :S ⊢Σ F ⊢Σ ∀x:S.F ⊢Σ ∀x:S.F ⊢Σ s : S ⊢Σ F[x/s] ⊢Σ s : S ⊢Σ F[x/s] ⊢Σ ∃x:S.F x :S, F ⊢Σ F ′ ∃x:S.F ⊢Σ F ′

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Example and Completeness Criterion

◮ Extending LF encodings to cover model theory surprisingly

simple

◮ Example: two axioms needed in the LF meta-logic to encode

first-order logic

◮ non-empty model universes: ∃x:univ.true ◮ consistency: ¬∃x:proof(false).true

◮ Completeness criterion: All provable existential quantifiers

have witnesses Question: When can this criterion be applied?

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